Question

Suppose an object is thrown with the same initial velocity and direction on Earth and on the Moon, where the acceleration due to gravity is one-sixth its value on Earth. How will vertical velocity, time of fight, maximum height, and horizontal distance change?

I would appreciate if they were numerical values like doubles or triples. Thank you

Answer 1

Let's call

g = acceleration on earth

g '= acceleration on the moon.

We know that:

`g'= (1)/(6)g`

1. The formula for vertical velocity on earth is:

`V_y = V_0sin(\theta) - gt`

On the moon, we have:

`V_y'= V_0sin(\theta) - (1)/(6)gt`

`V_y -V_y'= V_0sin(\theta) - gt - V_0sin(\theta) + (1)/(6)gt`

`Vy -Vy'= (1)/(6)gt -gt`

`Vy -Vy'= gt((1)/(6) -1)`

`Vy'= Vy + gt((5)/(6))`

The vertical speed on the moon is greater than on earth by a factor of gt (5/6)

2. The formula for the time of flight on earth is:

`t_v = 2(V_0sin(\theta))/(g)`

On the moon it is:

`t_v'= 2(V_0sin(\theta))/(((1)/(6)g))`

`t_v'= 12(V_0sin(\theta))/(g)`

`(t_v')/(t_v) = (12)/(2)`

`t_v'= 6t_v`

The time of flight on the moon is 6 times greater than on earth.

3. The maximum height on earth is:

`h = (V_0^2sin^2(\theta))/(2g)`

On the moon:

`h' = (V_0^2sin^2(\theta))/(2((1)/(6)g))\\\\h' = 3(V_0^2sin^2(\theta))/(g)`

So:

`(h')/(h) = ([3(V_0^2sin^2(\theta))/(g)])/([(V_0^2sin^2(\theta))/(2g)])`

`h'= 6h`

The maximum height on the moon is 6 times greater than on earth.

4. On earth, the horizontal distance traveled is equal to:

`r_x = V_0cos(\theta)t_v\\\\r_x'= V_0cos(\theta)(6t_v)\\\\(r_x')/(r_x) = 6\\\\r_x'= 6r_x\\\\`

The horizontal distance traveled on the moon is 6 times greater than that of the earth.

Answer 2

As we know that

Time of flight is given by

`T = (2vsin\theta)/(g)`

Vertical height

`H = (v^2sin^2\theta)/(2g)`

Horizontal range

`R = (v^2sin2\theta)/(g)`

vertical speed

`v_y = vsin\theta`

now we can see all factors except vertical speed depends on gravity inversely

So here we can say that

Range on moon will increase by 6 times

Maximum vertical height increased by 6 times

Time of flight increased by 6 times

as all above depends inversely on gravity

vertical speed remains unchanged as it is independent of gravity